Using a .23 build of stgit I could reproduce this every time just by running stg pull:

$ stg pull
Popping all applied patches ... done
Pulling from "origin"
Already up to date.
Error in atexit._run_exitfuncs:
Traceback (most recent call last):
  File "C:\src\Python27\lib\atexit.py", line 24, in _run_exitfuncs
    func(*targs, **kargs)
  File "C:\src\Python27\lib\site-packages\stgit\lib\git\repository.py", line 122, in _shutdown
    os.kill(self._proc.pid(), signal.SIGTERM)
WindowsError: [Error 5] Access is denied
Error in sys.exitfunc:
Traceback (most recent call last):
  File "C:\src\Python27\lib\atexit.py", line 24, in _run_exitfuncs
    func(*targs, **kargs)
  File "C:\src\Python27\lib\site-packages\stgit\lib\git\repository.py", line 122, in _shutdown
    os.kill(self._proc.pid(), signal.SIGTERM)
WindowsError: [Error 5] Access is denied

I fixed this with the following update:

    def _shutdown(self):
        if self._proc:
            self._proc.stdin.close()
            try:
                os.kill(self._proc.pid(), signal.SIGTERM)
            except OSError:
                pass
            self._proc.wait()

I believe what's happening here is that the PID is no longer valid and we're trying to kill an already-dead process.